Let us consider a problem where you are asked to find infimum of the set \[\{\int_0^{1}\sqrt{1+f'(x)^2}dx\}_{f\in S}\] where \(S\) is the set of all \(f\in C^1(\mathbb{R})\) with the property that \(f(0)=10\) and \(f(1)=0\). When we see integral and differential together, that should remind us the famous fundamental theorem of calculus, which says that ...
Let us check for limit/limsup/liminf of the sequence \(\frac{n}{10^{\lceil \log_{10}n \rceil}}\), where the notation \(\lceil x \rceil\) means the smallest integer greater than or equal to \(x\). For example, \(\lceil 0.1 \rceil=1, \lceil 0.9 \rceil=1, \lceil -1.2 \rceil=-1, \lceil -2.5 \rceil=-2\) To compute limit (to have a hope of computing), we need to know it converge (which we can check by checking it is Cauchy sequence). ...